The analytical chemist is frequently confronted with the problem of analyzing a sample to identify one or more properties of the sample, such as the octane number of a gasoline sample, or the protein content of a wheat sample. A closely related problem is the analysis of a sample to identify some or all of its constituents, and to determine their concentrations. The concentration of a given constituent can be regarded as a chemical property of the sample.
On many occasions, an instrument such as an absorption spectrophotometer is used to analyze the sample. However, instruments respond with signals (e.g. voltages or current), not reports of properties. Therefore a relationship has to be built that relates responses to properties before the instrument can be used to analyze samples. The process of building a model for the prediction of the properties of a sample from the instrument's responses is called calibration.
The advantages flowing from a successful model can be considerable. Imagine for example that it was desired to measure the concentration of a certain chemical species (analyte) in a chemical process stream. The most direct measurement technique would be to extract a sample from the process stream, and subject the sample to traditional chemical analysis techniques, to determine the analyte concentration. However, such an approach is cumbersome, and does not provide real time information. However, if a model can be constructed that relates the unknown concentration to absorbence of light at one or more wavelengths, a significant simplification would result. Concentration could then be estimated by placing a suitable optical absorbence measuring device in the process stream itself, thereby providing an efficient and real time technique for estimating the analyte's concentration.
An example of a calibration model is the classical calibration curve. The responses of an instrument are measured with a set of standard samples (calibration standards) of known analyte concentration. Such a set of calibration standards will be referred to herein as a calibration set. If a linear model is appropriate, then the response r measured by the instrument can be written EQU r=b.sub.1 c+b.sub.0 ( 1)
where c is the concentration of the analyte of interest, and the constants b.sub.1 and b.sub.0 are the model parameters. This is a univariate problem, in that the instrument response r is assumed to be a function of a single variable c. To calibrate the instrument, the instrument response r is measured for each sample in the calibration set, each such sample having a known concentration c. A linear least squares regression technique may then be used to determine the values of the constants b.sub.1 and b.sub.0 that provide the best fit of Equation (1) to the measured data. Once the model parameters are determined, concentrations of future samples may be estimated from the model, based upon the measured instrument response r.
Such univariate calibration requires that the instrument response be dependent only on the concentration of the analyte of interest. In order to fulfill this condition, the analytical chemist either separates the analyte from other constituents of the sample that interfere with the instrument response, or uses a highly selective instrument. Thus classical univariate calibration demands that the chemist make certain that there are no interfering species. If inadvertently an interfering constituent is present, there is no way to detect the error, much less to correct it.
Unfortunately, it is rarely possible to find univariate models that provide useful real world information. For example, it is generally the case that light absorbence at any given wavelength is affected by many chemical species that may be present, by turbidity, etc. In this more common case, the problem is multivariate, i.e., the measurable absorbence is a function of multiple variables, and multivariate calibration must be used to obtain reliable estimates of the property of interest. The use of multivariate calibration of analytical instruments is a rapidly growing field, primarily due to the development of so-called biased multivariate regression methods, such as principal component regression (PCR) and partial leased squares (PLS).
A standard multivariate calibration technique is illustrated in schematic form in FIGS. 1a and 1b. The technique begins with the preparation of a plurality of calibration samples 12. The calibration samples are formulated such that they are typical examples of the material, e.g., process stream, that the model will be used to analyze. One or more properties of interest for each calibration sample are then determined, as indicated in block 14, to produce "property" data representing such properties. In the ease of a physical property such as octane number, an analytic technique of accepted reliability and accuracy is used to measure the octane number for each calibration sample. In the ease where the properties are the concentrations of different chemical species, the concentrations may simply be recorded during the preparation of the samples. One then performs a plurality of measurements of each calibration sample, using analytical instrument 16, to produce "measurement" data. For example, for the ease in which the analytical instrument is a spectrophotometer, the absorbence of each sample is measured at a plurality of different wavelengths. Then, using any multivariate calibration method (e.g., principle component regression, partial least squares regression), one combines the property data produced in step 14 with the measurement data produced by analytical instrument 16, to obtain a model 20, i.e., a mathematical relationship, between the measured absorbence and the property or properties of interest.
Once model 20 has been created, it can be used to estimate the property or properties of an unknown sample, as shown in FIG. 1b. In particular, the unknown sample 22 is analyzed using analytical instrument 16 (the same instrument shown in FIG. 1a), to produce measurement data that is input to model 20. The model produces an estimate of the property or properties of the unknown sample.
While the above-described multivariate calibration technique is very powerful, a significant limitation of the technique has recently been recognized. This limitation is based upon the fact that no two analytical instruments are precisely identical to one another, even instruments of the same type coming from the same manufacturer. Thus a calibration model determined using a first instrument cannot generally be used for other instruments of the same type, without a significant loss of accuracy. The ideal way to avoid this problem would be to calibrate each individual instrument, using all of the samples in the calibration set. However, complete recalibration of each instrument is impractical in many situations. Complete recalibration, for example, could often require the transportation of a large number of calibration samples to the site of each instrument, a challenging task when the samples are numerous, chemically or physically unstable, or hazardous. It would therefore be highly useful to develop techniques for transferring a calibration model derived for a first, reference instrument for use on a second, target instrument.
A closely related problem occurs when the responses measured on a single instrument change over a period of time for any reason, for example, temperature fluctuations, electronic drift, wavelength or detector instability, etc. If such changes occur after the development of a calibration model, then subsequent use of the model may produce erroneous results. Conceptually, the problem of differences between instruments is nearly identical to the problem of variation of a single instrument over time, although they are associated with different causes. Both problems involve calibration on a reference instrument, and an attempt to use the calibration model on a target instrument that produces responses that differ from those of the reference instrument in some way. The target instrument may be either the same instrument at a later time, or a different instrument.
It is in the area of NIR (near infrared) analysis of agricultural products that the most work in transferring calibration models has been accomplished. Osborne and Fearn (1983) investigated the effects of transferring single wavelength calibration models between nine different instruments for the prediction of protein and moisture in wheat flour, using NIRA spectroscopy. Single wavelength bias correction terms for the two calibration equations on each instrument were determined, and the long-term stability of the calibration was studied. Later, Shenk, Westerhaus and Templeton (1985) published results from a study where a large number of candidate calibration equations were developed on a single instrument for the prediction of several properties related to the forage quality of grasses, and then transferred to six other instruments. The "best" equation was adjusted for bias, offset and wavelength selection on the other instruments, and the standard error of prediction was compared between the original and the other instruments for a set of 60 common samples.
Recently, Mark and Workman (1988) published work describing the selection of wavelengths for NIR calibration, based upon their robustness toward wavelength shifts between instruments. Unfortunately, all of the above-described methods use only a single, or small number, of wavelengths, and are not generally applicable to a multivariate calibration. U.S. Pat. No. 4,866,644 describes a calibration method that attempts to correct for full spectral responses. However, the applicability of the technique described in this patent is limited by its univariate nature.